It is well known that, if \(p(\cdot ) \in L H\left(\mathbb {R}^{d}\right)\)  with \(1<p_{-} \le p^{+}<\infty \)  the classical Hardy-Littlewood maximal function \(\mathcal {M}\)  is bounded on the variable Lebesgue space \(L^{p(\cdot )}(R^{d}),\)  see [1]}. However, it is known that even though these are the sharpest possible point-wise conditions, they are not necessary. In [2]} a necessary and sufficient condition is given for the \(L^{p(\cdot )}\) -boundedness of \(\mathcal {M},\)  but it is not an easy to work condition. The class \(L H(\mathbb {R}^{d})\)  is also sufficient for the boundedness on \(L^{p(\cdot )}\) -spaces of classical singular integrals of Calderón-Zygmund type, see [3]}.
